UT Austin charges international students the non-resident tuition rate, which runs far above the Texas resident rate. The Competitive Scholarship Waiver moves a recipient onto the resident rate for the terms they are enrolled, which is the single largest saving available to an international student at the university.
The waiver attaches to a scholarship rather than standing on its own. You need a competitive scholarship administered by a scholarship committee the university recognises, and only a limited number of waivers are issued each year. Departments and colleges hold most of these scholarships, so the department you apply to is the place to ask.
The paperwork goes through your department's undergraduate or graduate student service office, and the waiver has to be filed while you are enrolled. Plan on paying resident tuition and all fees yourself.